triangle calculator
Scientific CalculatorsEnter any 3 known values (sides or angles) to instantly calculate the rest.
How to Use the Triangle Calculator (Step-by-Step)
Solve any triangle with SSS, SAS, ASA, AAS, or SSA, inspect Heron's formula, verify inequalities, and calculate roof pitch.
Select Math or Construction / Roofing Mode
Choose 'Abstract Math' for standard geometry notation (sides a, b, c and angles A, B, C) or 'Construction / Roofing' for carpentry and truss framing (Span, Rise, Pitch).
Toggle Metric or Imperial Units
Switch between Metric (meters, cm, mm) and Imperial (in, ft, with builder fraction support such as 5 1/2") according to your project specifications.
Input Any 3 Known Parameters
Enter any three known values: three sides (SSS), two sides and an angle (SAS or SSA), or one side and two angles (ASA or AAS). At least one side must be specified.
Inspect the Interactive Proportional SVG Diagram
Observe the real-time dynamically rendered triangle schematic labeled with computed vertices, interior angles, and scaled side proportions.
Review Classification, Area & Perimeter
Examine computed surface area via Heron's formula, perimeter, semi-perimeter, and geometric classifications (Equilateral, Isosceles, Scalene, Acute, Right, Obtuse).
Expand Step-by-Step Proofs & Copy Dimensions
Click 'Show Math Steps' to view detailed algebraic substitutions for the Law of Cosines, Law of Sines, or Heron's theorem, and copy results to your clipboard.
Frequently Asked Questions (FAQ)
Triangle trigonometry, solving cases (SSS, SAS, ASA, SSA), Heron's formula, and construction roofing layout.
QWhat combinations of sides and angles are required to solve a triangle?
To solve an oblique triangle, you must know at least three parts, including at least one side: SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), or SSA (Side-Side-Angle). Knowing only three angles (AAA) is unsolvable because it determines triangle shape and proportions, but not absolute scale.
QWhat is the Triangle Inequality Theorem and why is it enforced?
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side (a + b > c, a + c > b, and b + c > a). If a + b ≤ c, the two shorter sides cannot meet to enclose an area, forming either a straight line segment or an impossible gap.
QHow does the Law of Cosines work and when is it applied?
The Law of Cosines relates all three sides to one angle: c² = a² + b² - 2ab·cos(C). It is used in SSS cases to solve for the largest angle via cos(C) = (a² + b² - c²) / (2ab), and in SAS cases to calculate the unknown third side opposite the known included angle.
QWhat is the ambiguous case in SSA (Side-Side-Angle) trigonometry?
When you know two sides and a non-included angle opposite one of them, the geometric altitude h = b·sin(A) dictates the outcome: if a < h, no triangle exists; if a = h, exactly one right triangle exists; if h < a < b, two distinct valid triangles can be formed (one acute, one obtuse); and if a ≥ b, exactly one triangle exists.
QWhat is Heron's Formula and how does it calculate area without height?
Heron's Formula calculates the area of any triangle given only its three sides (a, b, c). First compute the semi-perimeter s = (a + b + c) / 2. Then Area = √[s(s - a)(s - b)(s - c)]. It delivers exact area without needing perpendicular altitude lines or trigonometric functions.
QHow are triangles classified by sides and by interior angles?
By side lengths: Equilateral (all 3 sides equal, angles 60°), Isosceles (2 sides equal, 2 angles equal), and Scalene (all 3 sides unequal). By interior angles: Acute (all 3 angles < 90°), Right (one angle = 90°), and Obtuse (one angle > 90°).
QHow does Construction / Roofing mode calculate rafters and roof pitch?
In roofing mode, the Span is the total horizontal building width. The Half-Span is Span / 2. Rafter length (pitch) is calculated using the Pythagorean relation: Rafter = √[(Span/2)² + Rise²]. The pitch slope is often expressed as Rise over Run (e.g., 6/12 pitch means 6 inches of vertical rise for every 12 inches of horizontal run).
QWhy must the sum of interior angles in a triangle always equal 180°?
In Euclidean planar geometry, drawing a straight line parallel to the base through the opposite vertex forms alternate interior angles that lie on a straight line. Because a straight line measures 180°, the interior angles of any triangle must satisfy A + B + C = 180°.
- a, b, c: The three outer boundary sides of the triangle.
- A, B, C: The interior angles opposite to sides a, b, and c respectively.
- s (Semiperimeter): Calculated as (a + b + c) / 2.
What is triangle calculator?
| Interface Element | Mathematical Function | Practical Application |
|---|---|---|
| Construction / Roofing | Practical Mode | Optimizes the solver environment for professionals who need real-world architectural measurements. |
| Angle A/B/C (°) | Vertex Definition | Sets the exact incline of the internal corners of your geometric figure. |
| Copy Dimensions | Data Export | Copies the complete summary of sides, angles and area to paste into a document. |
| Show Math Steps | Educational Breakdown | Reveals the logical process behind the final calculation for total transparency. |